Let $f(x)=x^2-7 x$ and $g(x)=6+x$. Find the following.
(a) $(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d) $\left(\frac{f}{g}\right)(x)$
(e) The domain of $\frac{f}{g}
Answer (1)
Calculate ( f × g ) ( x ) by multiplying f ( x ) and g ( x ) , resulting in x 3 − x 2 − 42 x .
Determine ( g f ) ( x ) by dividing f ( x ) by g ( x ) , which simplifies to 6 + x x ( x − 7 ) .
Find the domain of ( g f ) ( x ) by ensuring the denominator 6 + x is not zero, thus x = − 6 .
State the final answers: ( f × g ) ( x ) = x 3 − x 2 − 42 x , ( g f ) ( x ) = 6 + x x ( x − 7 ) , and the domain of ( g f ) ( x ) is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 − 7 x and g ( x ) = 6 + x . We need to find ( f × g ) ( x ) , ( g f ) ( x ) , and the domain of ( g f ) ( x ) .
Calculating (f * g)(x) To find ( f × g ) ( x ) , we multiply the two functions: ( f × g ) ( x ) = f ( x ) × g ( x ) = ( x 2 − 7 x ) ( 6 + x ) Expanding this expression, we get: ( f × g ) ( x ) = x 2 ( 6 + x ) − 7 x ( 6 + x ) = 6 x 2 + x 3 − 42 x − 7 x 2 = x 3 − x 2 − 42 x So, ( f × g ) ( x ) = x 3 − x 2 − 42 x .
Calculating (f/g)(x) To find ( g f ) ( x ) , we divide f ( x ) by g ( x ) : ( g f ) ( x ) = g ( x ) f ( x ) = 6 + x x 2 − 7 x We can factor the numerator: ( g f ) ( x ) = 6 + x x ( x − 7 ) So, ( g f ) ( x ) = 6 + x x ( x − 7 ) .
Finding the Domain of (f/g)(x) To find the domain of ( g f ) ( x ) , we need to determine the values of x for which the denominator is not equal to zero. That is, we need to find the values of x such that g ( x ) = 6 + x = 0 . Solving for x , we get: 6 + x = 0 ⟹ x = − 6 Therefore, the domain of ( g f ) ( x ) is all real numbers except x = − 6 . In interval notation, this is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Final Answer In summary: (c) ( f × g ) ( x ) = x 3 − x 2 − 42 x (d) ( g f ) ( x ) = 6 + x x ( x − 7 ) (e) The domain of ( g f ) ( x ) is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Examples
Understanding function operations like multiplication and division is crucial in many real-world applications. For instance, if f ( x ) represents the area of a rectangular garden with variable side length x , and g ( x ) represents the cost per unit area depending on x , then ( f × g ) ( x ) gives the total cost of the garden. Similarly, ( g f ) ( x ) could represent the number of gardens you can afford given a fixed budget. Determining the domain ensures that the calculations are meaningful and realistic, avoiding scenarios where the cost per unit area becomes undefined.
